Sensation Perice&Jatrow

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Classics in the History of Psychology

An internet resource developed by Christopher D. Green

York University, Toronto, Ontario ISSN 1492-3713

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On Small Differences in Sensation

By Charles Sanders Peirce & Joseph Jastrow (1885)

First published in Memoirs of the National Academy of Sciences, 3, 73-83.

Presented 17 October 1884

Posted Jan 2005

Editor's note: Thanks to Joseph M. Ransdell of Texas Tech University for

providing me with an electronic version of this text. -cdg-

The physiological psychologists assume that two nerve excitations alike in

quality will only produce distinguishable sensations provided they differ in

intensity by an amount greater than a fixed ratio. The least perceptible difference

of the excitations divided by half their sum is what they call

the Unterschiedsschwelle. Fechner[1] gives an experiment to prove the fact

assumed, namely: He finds that two very dim lights placed nearly in line with the

edge of an opaque body show but one shadow of the edge. It will be found,

however, that this phenomenon is not a clearly marked one, unless the lights arenearly in range. If the experiment is performed with lateral shifting of one of the

lights, and with a knowledge of the effects of a telescope upon the appearance of

terrestrial objects at night, it will be found very far from conclusive.

The conception of the psychologists is certainly a difficult one to seize.

According to their own doctrine, in which the observed facts seem fully to bear

them out, the intensity of the sensation increases continuously with the

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excitation, so that the least increase of the latter must produce a corresponding

increase of the former. And, indeed, the hypothesis that a continuous increase of

the excitation would be accompanied by successive discrete increments of the

sensation, gratuitous as it would be, would not be sufficient to account for a

constant Unterschiedsschwelle. We are therefore forced to conclude that if there

be such a phenomenon, it has its origin, not in the faculty of sensation, but in thatof comparing sensations. In short, if the phenomenon were established, we

should be forced to say that there was a least perceptible difference of sensation -

- a difference which, though existing in sensation, could not be brought into

consciousness by any effort of attention. But the errors of our judgments in

comparing our sensations seem sufficiently accounted for by the slow and

doubtless complicated process by which the impression is conveyed from the

periphery to the brain; for this must be liable to more or less accidental

derangement at every step of its progress. Accordingly we find that the

frequencies of errors of different magnitudes follow the probability curve, which

is the law of an effect brought about by the sum of an infinite number of

infinitesimal causes. This theory, however, does not admit of

an Unterschiedsschwelle. On the contrary, it leads to the method of least squares,

according to which the multiplication of observations will indefinitely reduce the

error of their mean, so that if of two excitations one were ever so little the more

intense, in the long run it would be judged to be the more intense the majority of

times. It is true that the astronomers themselves have not usually supposed that

this would be the case, because (apart from constant errors, which have no

relevancy to the present question) they have supposed this extreme result to be

contrary to common sense. But it has seemed to us that the most satisfactorycourse would be to subject the question to the test of direct experiment. If there

be a least perceptible difference, then when two excitations differing by less than

this are presented to us, and we are asked to judge which is the greater, we ought

to answer wrong as often as right in the long run. Whereas, if the theory of least

squares is correct, we not [p. 76] only ought to answer right oftener than wrong,

but we ought to do so in a predictable ratio of cases.[2]

We have experimented with the pressure sense, observing the proportion of

errors among judgments as to which is the greater of two pressures, when it is

known that the two are two stated pressures, and the question presented for thedecision of the observer is, which is which? From the probability, thus

ascertained, of committing an error of a given magnitude, the probable error of a

judgment can be calculated according to the mathematical theory of errors. If,

now, we find that when the ratio of the two pressures is smaller than a certain

ratio, the erroneous judgments number one-half of the whole, while the

mathematical theory requires them to be sensibly fewer, then this theory is







plainly disproved, and the maximum ratio at which this phenomenon is observed

the so-called Unterschiedsschwelle. If, on the other hand, the values obtained for

the probable error are the same for errors varying from three times to one-fourth

of the probable error (the smallest for which it is easy to collect sufficient

observations), then the theory of the method of least squares is shown to hold

good within those limits, the presumption will be that it extends still further, andit is possible that it holds for the smallest differences of excitation. But, further, if

this law is shown to hold good for difference so slight that the observer is not

conscious of being able to discriminate between the sensations at all, all reason

for believing in an Unterschiedsschwelleis destroyed. The mathematical theory

has the advantage of yielding conceptions of greater definiteness than that of the

physiologists, and will thus tend to improve methods of observation. Moreover, it

affords a ready method for determining the sensibility or fineness of perception

and allows of a comparison of one observer's results with the results of others;

for, knowing the number of errors in a certain number of experiments, and

accepting the conclusions of this paper, the calculated ratio to the total excitation

of that variation of excitation, in judging which we should err one time out of

four, measures the sensibility. Incidentally our experiments will afford additional

information upon the value of the normal average sensibility for the pressure

sense, which they seem to make a finer sense than it has hitherto been believed to

be. But in this regard two things have to be noted: (1) Our value relates to the

probable error or the value for the point at which an error is committed half the

time; (2) in our experiments there were two opportunities for judging, for the

initial weight was either first increased and then diminished, or vice versa, the

subject having to say which of these two double changes was made. It wouldseem at first blush that the value thus obtained ought to be multiplied by

√2(1.414) to get the error of a single judgment. Yet this would hardly be correct,because the judgment, in point of fact, depended almost exclusively on the

sensation of increase of pressure, the decrease being felt very much less. The

ratio √2(1.414) would therefore be too great, and 1.2 would perhaps be aboutcorrect. The advantage of having two changes in one experiment consists in this:

If only one change were employed, then some of the experiments would have an

increase of excitation only and the others a decrease only; and since the former

would yield a far greater amount of sensation than the latter, the nature of the

results would be greatly complicated; but when each experiment embraces a [p.77]double change this difference in the amount of sensation caused by an

increase and decrease of pressure affects every experiment alike, and the liability

to error is constant.[3]

Throughout our observations we noted the degree of confidence with which the

observer gave his judgment upon a scale of four degrees, as follows:







0. denoted absence of any preference for one answer over its opposite, so

that it seemed nonsensical to answer at all.

1. denoted a distinct leaning to one alternative.

2. denoted some little confidence of being right.

3. denoted as strong a confidence as one would have about such sensations.

We do not mean to say that when zero was the recorded confidence, there was

absolutely no sensation of preference for the answer given. We only mean that

there was no sensation that the observer noticed when attending to his feelings of

this sort as closely as he conveniently could, namely, closely enough to mark

them on this scale. The scale of confidence fluctuated considerably. Thus, when

Mr. Jastrow passed from experiments upon differences of weight of 60, 30, and

15 on the thousand to differences of 20, 10, and 5 on the thousand, although the

accuracy of his judgments was decidedly improved, his confidence fell off very

greatly, owing to his no longer having the sensation produced by a difference of

60 present to his memory. The estimations of confidence were also rough, andmight be improved in future work. The average marks seem to conform to the

formula--

m = c log ( p /1- p) where m denotes the degree of confidence on the scale, p denotes the probability

of the answer being right, and c is a constant which may be called the index of

confidence.

To show that this formula approximates to the truth, we compare it with theaverage marks assigned to estimates of differences for which more than a

hundred experiments were made. Mr. Jastrow's experiments are separated into

groups, which will be explained below.



[p. 78]

The judgments enunciated with any given degree of confidence were more likelyto be right with greater differences than with smaller differences. To show this,

we give the frequency of the different marks in Mr. Jastrow's second, third, and

fourth groups.[4]

[ Editor's note: The table immediately below is misplaced within footnote 4 in the

original publication. -cdg-]







The apparatus used was an adaptation of a "Fairbanks" post-office scale; upon

the end of the beam of which was fixed a square enlargement (about one-half

inch square), with a flat top, which served to convey the pressure to the finger in

a manner to be presently described. This was tightly covered with an India-

rubber cap, to prevent sensations of cold, etc., from contact with the metal. A

kilogram placed in the pan of the balance brought a pressure of one-fourth of itsweight upon the finger. The differential pressure was produced by lowering upon

the pan of the balance a smaller pan into which the proper weights could be

firmly fixed; this little pan had its bottom of cork, and was placed upon a piece of

flannel which constantly remained in the pan of the balance. It was lifted off and

on by means of a fine India-rubber thread, which was so much stretched by the

weight as certainly to avoid any noise or jar from the momentum of the

descending pan. A sufficient weight could also be hung on the beam of the



balance, so as to take off the entire pressure from the finger at the end of each

experiment. This weight could be applied or removed by means of a cam acting

upon a lever; and its bearings upon the beam were guarded by India-rubber. It

was found that the use of this arrangement, which removed all annoying

irregularities of sensation connected with the removal and replacement of the

greater (initial) pressure, rendered the results more uniform and diminished theprobable error. It also shortened the time necessary for performing the

experiments, so that a series of 25 experiments was concluded before the effects

of fatigue were noticeable. It may be mentioned that certain causes tended to the

constant decrease of the probable error as the experiments went on, these mainly

being an increased skill on the part of the operator and an education of the

sensibility of the subject. The finger was supported in such a way as to be lightly

but firmly held in position, all the muscles of the arm being relaxed; and the

India-rubber top of the brass enlargement at the end of the beam of the balance

was never actually separated from the finger. The projecting arm of a filter-stand

(the height of which could be adjusted) with some attachments not necessary to

detail, gently prevented the finger from moving upwards under the pressure

exerted by the weight in the pan. In the case of Mr. Peirce as subject (it may be

noted that Mr. Peirce is left-handed, while Mr. Jastrow is strongly right-handed)

the tip of forefinger, and in the case of Mr. Jastrow of the middle finger, of the

left hand were used. In addition, a screen served to prevent the subject from

having any indications whatever of the movements of the operator. It is hardly

necessary to say that we were fully on guard against unconsciously received

indications.

The observations were conducted in the following manner: At each sitting three

differential weights were employed. At first we always began and ended with the

heaviest, but at a later period the plan was to begin on alternate days with the

lightest and heaviest. When we began with the heaviest 25 observations [ 5] were

made with that; then 25 with the middle one, and then 25 with the lightest; this

constituted one-half of the sitting. It was completed by three more sets of 25, the

order of the weights being reversed. When we began with the lightest the

heaviest was used for the third and fourth sets. In this way 150 experiments on

each of us were taken at one sitting of two hours.

A pack of 25 cards were taken, 12 red and 13 black, or vice versa, so that in the

50 experiments made at one sitting with a given differential weight, 25 red and

25 black cards should be used. These cards were cut exactly square and their

corners were distinguished by holes punched in them so as to indicate the scale of

numbers (0, 1, 2, 3) used to designate the degree of confidence of the judgment.

The backs of these cards were distinguished from their faces. They were, in fact,

made of ordinary playing-cards. At the beginning of a set of 25, the pack was






well shuffled, and, the operator and subject having taken their places, the

operator was governed by the color [p. 80] of the successive cards in choosing

whether he should first diminish the weight and then increase it, or vice versa. If

the weight was to be first increased and then diminished the operator brought the

pressure exerted by the kilogram alone upon the finger of the subject by means of

the lever and cam mentioned above, and when the subject said "change" hegently lowered the differential weight, resting in the small pan, upon the pan of

the balance. The subject, having appreciated the sensation, again said "change,"

whereupon the operator removed the differential weight. If, on the other hand,

the color of the card directed the weight to be first diminished and then increased,

the operator had the differential weight already on the pan of the balance before

the pressure was brought to bear on the finger, and made the reverse changes at

the command of the subject. The subject then stated his judgment and also his

degree of confidence, whereupon the total pressure was at once removed by the

cam, and the card that had been used to direct the change was placed face down

or face up according as the answer was right or wrong, and with corner indicating

the degree of confidence in a determinate position. By means of these trifling

devices the important object of rapidity was secured, and any possible

psychological guessing of what change the operator was likely to select was

avoided. A slight disadvantage in this mode of proceeding arises from the long

runs of one particular kind of change, which would occasionally be produced by

chance and would tend to confuse the mind of the subject. But it seems clear that

this disadvantage was less than that which would have been occasioned by his

knowing that there would be no such long runs if any means had been taken to

prevent them. At the end of each set the results were of course entered into abook.[6]

The following tables show the results of the observations for each day:






The numbers in the columns show the number of errors in fifty experiments.

With the average number of errors in a set of fifty we compare the theoretical

value of this average as calculated by the method of least squares. The number

.051 thus obtained in this case best satisfies the mean number of errors. The

numbers affixed with a sign denote, in the upper row the observed ( a posteriori)probable error of the mean value as given, in the lower row the calculated (a

priori) probable error. The last two lines give the average confidence observed

and calculated with each variation of the ratios of pressure. It will be seen that the

correspondence between the real and theoretical numbers is close, and closest

when the number of sets is large. The probable errors also closely correspond, the

observed being, as is natural, slightly larger than the calculated probable

errors.[p. 81]



The following is a similar table for Mr. Jastrow as subject:It would obviously be

unfair to compare these numbers with any set of theoretical numbers, since the

probable error is on the decrease throughout, owing to effects of practice, etc. For

various reasons we can conveniently group these experiments into four groups.

The first will include the experiments from December 10 to January 22,

inclusive; the second from January 24 to February 24, inclusive; the third from

March 4 to March 25, inclusive; the fourth from March 30 to the end of the

work. The mean results for the different groups are exhibited in the following

tables:[p. 82]





The tables show that the numbers of errors follow, as far as we can conveniently

trace them, the numbers assigned by the probability curve,[7] and therefore

destroy all presumption in favor of anUnterschiedsschwelle. The introduction and

retention of this false notion can only confuse thought, while the conception of

the mathematician must exercise a favorable influence on psychological

experimentation.[8]

The quantity which we have called the degree of confidence was probably the

secondary sensation of a difference between the primary sensations compared.

The evidence of our experiments [p. 83] seems clearly to be that this sensation

has no Schwelle, and vanishes only when the difference to which it refers

vanishes. At the same time we found the subject often overlooked this element of

his field of sensation, although his attention was directed with a certain strength

toward it, so that he marked his confidence as zero. This happened in cases where

the judgments were so much affected by the difference of pressures as to be

correct three times out of five. The general fact has highly important practical

bearings, since it gives new reason for believing that we gather what is passing in

one another's minds in large measure from sensations so faint that we are not

fairly aware of having them, and can give no account of how we reach our

conclusions about such matters. The insight of females as well as certain

"telepathic" phenomena may be explained in this way. Such faint sensations










ought to be fully studied by the psychologist and assiduously cultivated by every

man.

Notes [1] Elemente der Psychophysik, I, p. 242.

[2] The rule for finding this ratio is as follows: Divide the logarithm of the ratio

of excitations by the probable error and multiply the quotient by 0.477. Call this

product t. Enter it in the table of the integral θt, given in most works on

probabilities; θt is the proportion of cases in which the error will be less than the

difference between the given excitations. In all these cases, of course, we shall

answer correctly, and also by chance in one-half of the remaining cases. The

proportion of erroneous answers is therefore (1-θt)/2. In the following table the

first column gives the quotient of the logarithm of the ratio of excitation, divided

by the probable error, and the second column shows the proportion of erroneous

judgments:To guess the correct card out of a pack of fifty-two once in eleven

times it would be necessary to have a sensation amounting to 0.37 of the

probable error. This would be a sensation of which we should probably never

become aware, as will appear below.

[3] The number of errors, when an increase of weight was followed by a

decrease, was slightly less than when the first change was a decrease of pressure.



[4] The result of our observations on the confidence connected with the judgments is as follows:

In 1,125 experiments (subject, Mr. Peirce) -- variations 15, 30, and 60 grams --

there occurred confidence of 3, 35 times (3 per cent.); of 2, 102 times (9 per

cent.); of 1, 282 times (25 per cent.); of 0, 706 times (63 per cent.). In these

experiments there were 332 (29 per cent.) errors committed, of which 1 (0.3 per

cent.) was made in connection with a confidence 3; 10 (3 per cent.) with a

confidence 2; 51 (15 per cent.) with a confidence 1; 270 (81 per cent.) with a

confidence 0. From which we find that in connection with a confidence of 3 there

occurred 1 error in 35 cases (3 per cent.); with a confidence of 2, 10 errors in 102

cases (10 per cent.); with a confidence of 1, 51 errors in 282 cases (18 per cent.);

with a confidence of 0, 270 errors in 706 cases (38 per cent.).

In 1,975 experiments (subject, Mr. Jastrow) -- variations 15, 30, and 60 grams --

there occurred confidence of 3, 62 times (3 per cent.); of 2, 196 times (10 per

cent.); of 1, 594 times (30 per cent.); of 0, 1,123 times (57 per cent.). In these

experiments there were 451 (23 per cent.) errors committed, of which 2 (0.4 per

cent.) were made in connection with a confidence of 3; 12 (3 per cent.) with a

confidence of 2; 97 (22 per cent.) with a confidence of 1; 340 (75 per cent.) with

a confidence of 0. Again, in connection with a confidence of 3, errors occurred

twice in 62 cases (3 per cent.); with a confidence of 2, 12 times in 196 cases (6

per cent.); with a confidence of 1, 97 times in 504 cases (16 per cent.); with a

confidence of 0, 340 times in 1,123 cases (30 per cent.).

In 1,675 experiments (subject, Mr. Jastrow) -- variations 5, 10, and 20 grams --

there occurred confidences of 3, none; of 2, none; of 1, 115 times (7 per cent.); of

0, 1,560 times (93 per cent.). In these experiments there were 538 (32 per cent.)



errors committed, of which 16 (3 per cent.) occurred in connection with a

confidence of 1; 522 (97 per cent.) with a confidence of 0. Again, in connection

with a confidence of 1, errors occurred 16 times in 115 cases (14 per cent.); with

a confidence of 0, 522 times in 1,560 cases (34 per cent.).

[5] At first a short pause was made in the set of 25, at the option of the subject;later this was dispensed with.

[6] In the experiments of December, 1883, and January, 1884, the method as

above described was not fully perfected, the most important fault being that the

total weight instead of being removed and replaced by a mechanical device, was

taken off by the operator pressing with his finger upon the beam of the balance.

[7] In the tables of the third and fourth groups, there is a markeddivergence

between the a priori and a posteriori probable error, for theaverage number of

errors in 50, making the observed probable error toosmall. This can only bepartly accounted for by the fact that thesubject formed the unconscious habit of

retaining the number of eachkind of experiment in a set and answering according

to that knowledge.In point of fact the plus errors and minus errors separately do

notexhibit the singular uniformity of their sums, for which we are quiteunable to

account. Thus in the fourth group we have:[8] The conclusions of this paper are

strengthened by the results of a series of experiments on the color sense, made



with the use of a photometer by Mr. Jastrow. The object was to determine the

number of errors of a given magnitude, and compare the numbers thus

ascertained with the theoretical numbers given by the probability curve. A

thousand experiments were made. Dividing the magnitude of the errors from 0 to

the largest error, made into 5 parts, the number of errors, as observed and

calculated, that occur in each part are as follows: These numbers would be incloser accordance if the probable error were the same throughout, as it is not

owing to the effects of practice, etc. Moreover, the experiments were made on

different colors -- 300 on white and 100 each on yellow, blue, dove, pink, green,

orange, and brown. These experiments were not continuous.

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